Numerical analysis of Hahn echoes in solids

A numerical procedure for computing the Hahn echoes of half-integer quadrupole spins (I=3/2, 5/2, 7/2, and 9/2) in solids has been derived from a detailed analysis of the evolution of the density operator. As the first-order quadrupole interaction is taken into account throughout the experiment, consisting in exciting the spin system with two in-phase pulses, the results are valid for any ratio of the quadrupole coupling, ${\mathrm{\omega }}_{\mathit{Q}}$, to the amplitude of the pulses, ${\mathrm{\omega }}_{\mathrm{RF}}$. In the hard-pulse excitation condition (${\mathrm{\omega }}_{\mathit{Q}}$/${\mathrm{\omega }}_{\mathrm{RF}}$≪1), the central-transition echo reaches the maximum when the first- and the second-pulse flip angles are equal to π/2 and π, respectively. On the other hand, all the echoes should be observed if the second-pulse flip angle is smaller than π/2; the optimum value of this angle depends on the spin I. The results on the central-transition echo in the soft-pulse excitation condition (${\mathrm{\omega }}_{\mathit{Q}}$/${\mathrm{\omega }}_{\mathrm{RF}}$≤50), are presented. The two pulse flip angles should be optimized in order to observe the echo. Moreover, the study of the amplitude of the central-transition echo as a function of the second-pulse flip angle makes it possible to determine the value of ${\mathrm{\omega }}_{\mathit{Q}}$ in a single crystal, or those of the quadrupole coupling constant and the asymmetry parameter in a powder.